My friend gave me a problem, which can be reduced to the following.

Let $$$S(k)$$$ be the set of all arrays of size $$$n$$$ that contains $$$k$$$ ones and $$$n - k $$$ zeroes. Let for some array $$$s \in S(k)$$$ , $$$pos_s[1..k]$$$ denotes the position of those ones (say in increasing order, it doesn't matter actually). We have to calculate.

i.e, for all arrays $$$s \in S(k)$$$ we have to sum the positions of all $$$k$$$ ones.

My thoughts : hmm, what about considering each position, but then there will be intersections, like I might put two ones at the same index, so ? what about inclusion-exclusion? No it's getting a lot messy.

Linearity of Expectation : Let's find the expected value of $$$ \sum_{i = 1}^{k} pos_s[i] $$$

$$ E[\sum_{i = 1}^{k} pos_s[i]